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Algebra Qualifying Exam September 13, 2012 Do all five problems. 1. Let G be a group. A subgroup H is said to be normal if gHg −1 = H for every g ∈ G. Show that it is enough to show that gHg −1 ⊆ H for every g ∈ G. 2. If H and K are normal subgroups of a group G with HK = G, prove that G/(H ∩ K) ∼ = (G/H) × (G/K). Note that you may apply the First Isomorphism Theorem but not the Second or Third. 3. Let R be an integral domain (with multiplicative identity 1 6= 0) and let Q be its field of fractions. [Recall that Q is defined to be set of equivalence classes ab where a ∈ R, b ∈ R \ {0}; here the equivalence relation is defined so that ab = dc if and only if ad = bc and addition and multiplication are defined as usual for fractions.] Prove: if σ : R → R is an automorphism then there is a unique automorphism σ : Q → Q such that σ r 1 = σ(r) for all r ∈ R. 1 4. Let V be a finite-dimensional vector space and let T : V → V be linear. Prove that if rank(T ) = rank(T 2 ), then V = range (T ) ⊕ null (T ). 5. Let V denote the vector space (over R) of polynomials of degree ≤ 2 with real coefficients. Define T : V → V by T (p)(x) = p(x) + 3xp0 (x) + 4p00 (x). Then (a) Find [T ]β , the matrix representation with respect to the basis β = (1, x, x2 ). (b) Is the transformation T diagonalizable? If so express [T ]β as a product QDQ−1 where D is a diagonal matrix, otherwise explain why T is not diagonalizable.